Asymptotic results for certain first-passage times and areas of renewal processes
Authors:
Claudio Macci and Barbara Pacchiarotti
Journal:
Theor. Probability and Math. Statist. 108 (2023), 127-148
MSC (2020):
Primary 60F10, 60F05, 60K05
DOI:
https://doi.org/10.1090/tpms/1189
Published electronically:
May 2, 2023
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Additional Information
Abstract: We consider the process $\{x-N(t):t\geq 0\}$, where $x\in \mathbb {R}_+$ and $\{N(t):t\geq 0\}$ is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of $(\tau (x),A(x))$ where $\tau (x)$ is the first-passage time of $\{x-N(t):t\geq 0\}$ to reach zero or a negative value, and $A(x)≔\int _0^{\tau (x)}(x-N(t))dt$ is the corresponding first-passage (positive) area swept out by the process $\{x-N(t):t\geq 0\}$. We remark that we can define the sequence $\{(\tau (n),A(n)):n\geq 1\}$ by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as $x\to \infty$ in the fashion of large (and moderate) deviations.
References
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References
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- M. Abundo, D. Del Vescovo, On the joint distribution of first-passage time and first-passage area of drifted Brownian motion, Methodol. Comput. Appl. Probab. 19 (2017), no. 3, 985–996. MR 3683981
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Additional Information
Claudio Macci
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy
Email:
macci@mat.uniroma2.it
Barbara Pacchiarotti
Affiliation:
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy
Email:
pacchiar@mat.uniroma2.it
Keywords:
Large deviations,
moderate deviations,
joint distribution,
integrated random walk
Received by editor(s):
October 2, 2021
Accepted for publication:
April 5, 2022
Published electronically:
May 2, 2023
Additional Notes:
The authors acknowledge the support of GNAMPA-INdAM and of MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP E83C18000100006).
Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv